$$\begin{gathered} \text{Let}\mathrm{ Z}\text{ be the complex number satisfying}|z-5|\le 3\text{and having maximum positive argument, then}34\left|\frac{5z-12}{5z+16}\right|{}^2\text{is equal to} \\ \end{gathered}$$

The correct answer is 20.

Step 1: Understand the geometric locus of complex number z
The given inequality is:

|z-5|\le 3

This equation represents a closed disk in the complex plane centered at C = (5, 0) with a radius of r = 3.

Step 2: Find the complex number z with maximum positive argument
For z to have the maximum positive argument (or angle with the positive real axis), the line connecting the origin O(0,0) to z must be tangent to the circle |z - 5| = 3 in the upper half-plane.

Let the point of tangency be P representing z.

In the right-angled triangle OPC:
- The hypotenuse OC = 5 (distance from origin to center)
- The leg PC = 3 (radius of the circle)
- The leg OP = \sqrt{5^2-3^2}=4

If θ is the argument of z, then from right triangle OPC:

\cos \theta =\frac{4}{5},\sin \theta =\frac{3}{5}

Since the magnitude of z is |z| = OP = 4, we write z in polar form:

z=4(\cos \theta +i\sin \theta )=4(\frac{4}{5}+i\frac{3}{5})=\frac{16+12i}{5}

Multiplying both sides by 5 gives:

5z=16+12i

Step 3: Evaluate the required expression
Now we calculate the numerator and denominator terms of the expression:

Numerator:
5z-12=(16+12i)-12=4+12i

Denominator:
5iz+16=i(5z)+16=i(16+12i)+16=16i-12+16=4+16i

Now, calculate the magnitudes squared:

|5z-12{|}^2=4^2+{12}^2=16+144=160

|5iz+16{|}^2=4^2+{16}^2=16+256=272

Substitute these values back into the expression:

34{\left|\frac{5z-12}{5iz+16}\right|}^2=34\times \frac{160}{272}

Simplifying the fraction:

\frac{160}{272}=\frac{10}{17}

Thus:

34\times \frac{10}{17}=2\times 10=20

Therefore, the final answer is 20.